Fermionic linear optics is a model of quantum computation which is efficiently simulable on a classical probabilistic computer. We study the problem of a classical simulation of fermionic linear optics augmented with noisy auxiliary states. If the auxiliary state can be expressed as a convex combination of pure fermionic Gaussian states, the corresponding computation scheme is classically simulable. We present an analytic characterization of the set of convex-Gaussian states in the first nontrivial case, in which the Hilbert space of the ancilla is a four-mode Fock space. We use our result to solve an open problem recently posed by de Melo et al. [New J. Phys. 15, 013015 (2013)] and to study in detail the geometrical properties of the set of convex-Gaussian states.
Fermionic linear optics is a model of quantum computation which is efficiently simulable on a classical probabilistic computer. We study the problem of a classical simulation of fermionic linear optics augmented with noisy auxiliary states. If the auxiliary state can be expressed as a convex combination of pure fermionic Gaussian states, the corresponding computation scheme is classically simulable. We present an analytic characterization of the set of convex-Gaussian states in the first nontrivial case, in which the Hilbert space of the ancilla is a four-mode Fock space. We use our result to solve an open problem recently posed by de Melo et al. [New J. Phys. 15, 013015 (2013)] and to study in detail the geometrical properties of the set of convex-Gaussian states.
We consider the realization of universal quantum computation through braiding of Majorana fermions supplemented by unprotected preparation of noisy ancillae. It has been shown by Bravyi (2006 Phys. Rev. A 73 042313) that under the assumption of perfect braiding operations, universal quantum computation is possible if the noise rate on a particular four-fermion ancilla is below 40%. We show that beyond a noise rate of 89% on this ancilla the quantum computation can be efficiently simulated classically: we explicitly show that the noisy ancilla is a convex mixture of Gaussian fermionic states in this region, while for noise rates below 53% we prove that the state is not a mixture of Gaussian states. These results are obtained by generalizing concepts in entanglement theory to the setting of Gaussian states and their convex mixtures. In particular, we develop a complete set of criteria, namely the existence of a Gaussian-symmetric extension, which determine whether a state is a convex mixture of Gaussian states.
We propose efficient algorithms for classically simulating fermionic linear optics operations applied to non-Gaussian initial states. By gadget constructions, this provides algorithms for fermionic linear optics with non-Gaussian operations. We argue that this problem is analogous to that of simulating Clifford circuits with non-stabilizer initial states: Algorithms for the latter problem immediately translate to the fermionic setting. Our construction is based on an extension of the covariance matrix formalism which permits to efficiently track relative phases in superpositions of Gaussian states. It yields simulation algorithms with polynomial complexity in the number of fermions, the desired accuracy, and certain quantities capturing the degree of non-Gaussianity of the initial state. We study one such quantity, the fermionic Gaussian extent, and show that it is multiplicative on tensor products when the so-called fermionic Gaussian fidelity is. We establish this property for the tensor product of two arbitrary pure states of four fermions with positive parity.
Abstract This work explores displaced fermionic Gaussian operators with nonzero linear terms. We first demonstrate equivalence between several characterizations of displaced Gaussian states. We also provide an efficient classical simulation protocol for displaced Gaussian circuits and demonstrate their computational equivalence to circuits composed of nearest-neighbor matchgates augmented by single-qubit gates on the initial line. Finally, we construct a novel Gaussianity-preserving unitary embedding that maps n-qubit displaced Gaussian states to ( n + 1 ) -qubit even Gaussian states. This embedding facilitates the generalization of existing Gaussian testing protocols to displaced Gaussian states and unitaries. Our results provide new tools to analyze fermionic systems beyond the constraints of parity super-selection, extending the theoretical understanding and practical simulation of fermionic quantum computation.
We revisit the problem of learning fermionic linear optics (FLO), also known as fermionic Gaussian unitaries. Given black-box query access to an unknown FLO, previous proposals required O(n5/ε2) queries, where n is the system size and ε is the error in diamond distance. These algorithms also use unphysical operations (i.e., violating fermionic superselection rules) and/or n auxiliary modes to prepare Choi states of the FLO. In this work, we establish efficient and experimentally friendly protocols that obey superselection, use minimal ancilla (at most 1 extra mode), and exhibit improved dependence on both parameters n and ε. For arbitrary (active) FLOs this algorithm makes at most O(n4/ε) queries, while for number-conserving (passive) FLOs we show that O(n3/ε) queries suffice. The complexity of the active case can be further reduced to O(n3/ε) at the cost of using n ancilla. This marks the first FLO learning algorithm that attains Heisenberg scaling in precision. As a side result, we also demonstrate an improved copy complexity of O(nη2/ε2) for time-efficient state tomography of η-particle Slater determinants in ε trace distance, which may be of independent interest.
Fermionic linear optics is a limited form of quantum computation which is known to be efficiently simulable on a classical computer. We revisit and extend this result by enlarging the set of available computational gates: in addition to unitaries and measurements, we allow dissipative evolution governed by a Markovian master equation with linear Lindblad operators. We show that this more general form of fermionic computation is also simulable efficiently by classical means. Given a system of N fermionic modes, our algorithm simulates any such gate in time O(N3) while a single-mode measurement is simulated in time O(N2). The steady state of the Lindblad equation can be computed in time O(N3).
Fermionic linear optics is a limited form of quantum computation which is known to be efficiently simulable on a classical computer. We revisit and extend this result by enlarging the set of available computational gates: in addition to unitaries and measurements, we allow dissipative evolution governed by a Markovian master equation with linear Lindblad operators. We show that this more general form of fermionic computation is also simulable efficiently by classical means. Given a system of N~fermionic modes, our algorithm simulates any such gate in time O(N3) while a single-mode measurement is simulated in time O(N2). The steady state of the Lindblad equation can be computed in time O(N3).